Enhanced transverse magnetic fluctuations observed below the proton cyclotron frequency in the terrestrial magnetosheath have been identified as due to the proton cyclotron and helium cyclotron instabilities driven by the T⊥>T∥ condition of the sheath ions. One-dimensional hybrid computer simulations are used here to examine the nonlinear properties of these two growing modes at relatively weak fluctuation energies and for wave vectors parallel to the background magnetic field. Second-order theory predicts fluctuating magnetic field energies at saturation of the proton cyclotron anisotropy instability in semiquantitative agreement with the simulation results. In particular, introduction of the helium component enhances the wave-particle exchange rate for proton anisotropy reduction by that instability, thereby reducing the saturation energy of that mode. The simulations demonstrate that wave-particle interactions by the proton cyclotron and helium cyclotron instabilities lead to the anticorrelation of T⊥p/T∥p and &bgr;∥p, a result observed by Anderson and Fuselier (1993). ¿ American Geophysical Union 1993